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Geometric Correction

Lecture 5


Feb 18, 2005

1. What and why


Remotely sensed imagery typically exhibits
internal

and
external

geometric

error
.

It is important to recognize the
source of the internal and external error and whether it is
systematic

(predictable)

or

nonsystematic

(random). Systematic
geometric error is generally easier to identify and correct than
random geometric error


It is usually necessary to
preprocess

remotely sensed data and
remove geometric distortion so that individual picture elements
(pixels) are in their proper
planimetric

(
x,

y
) map locations.


This allows remote sensing

derived information to be related
to other thematic information in geographic information
systems (GIS) or spatial decision support systems (SDSS).


Geometrically corrected imagery

can be used to extract
accurate distance, polygon area, and direction (bearing)
information.

Internal geometric errors



Internal geometric errors

are introduced by the remote
sensing
system

itself or in combination with
Earth rotation

or
curvature characteristics
. These distortions are often
systematic

(predictable) and may be identified and corrected
using pre
-
launch or in
-
flight platform ephemeris (i.e.,
information about the geometric characteristics of the sensor
system and the Earth at the time of data acquisition). These
distortions in imagery that can sometimes be corrected
through analysis of sensor characteristics and ephemeris data
include:



skew
caused by Earth rotation effects
,



scanning system

induced

variation in ground resolution cell size,



scanning system

one
-
dimensional relief displacement
, and



scanning system

tangential scale distortion
.

External geometric errors



External geometric errors

are usually introduced by
phenomena that vary in nature through space and
time. The most important external variables that can
cause geometric error in remote sensor data are
random movements by the aircraft (or spacecraft) at
the exact time of data collection, which usually
involve:




altitude

changes, and/or



attitude

changes (roll, pitch, and yaw).

Altitude Changes



Remote sensing systems flown at a constant
altitude above ground level (AGL) result in
imagery with a uniform scale all along the
flightline. For example, a camera with a 12
-
in.
focal length lens flown at 20,000 ft. AGL will
yield 1:20,000
-
scale imagery. If the aircraft or
spacecraft gradually changes its altitude along a
flightline, then the scale of the imagery will
change. Increasing the altitude will result in
smaller
-
scale imagery (e.g., 1:25,000
-
scale).
Decreasing the altitude of the sensor system will
result in larger
-
scale imagery (e.g, 1:15,000). The
same relationship holds true for digital remote
sensing systems collecting imagery on a pixel by
pixel basis.


Attitude changes



Satellite platforms are usually stable because they are not
buffeted by atmospheric turbulence or wind. Conversely,
suborbital aircraft must constantly contend with atmospheric
updrafts, downdrafts, head
-
winds, tail
-
winds, and cross
-
winds
when collecting remote sensor data. Even when the remote
sensing platform maintains a constant altitude AGL, it may
rotate randomly about three separate axes that are commonly
referred to as
roll
,
pitch
, and
yaw.


Quality remote sensing systems often have
gyro
-
stabilization
equipment

that isolates the sensor system from the roll and
pitch movements of the aircraft. Systems without stabilization
equipment introduce some geometric error into the remote
sensing dataset through variations in roll, pitch, and yaw that
can only be corrected using
ground control points
.


2. Conceptions of geometric
correction


Geocoding
: geographical referencing


Registration:

geographically or nongeographically (no coordination system)


Image to Map (or Ground Geocorrection)


The correction of digital images to ground coordinates using ground control
points collected from maps (Topographic map, DLG) or ground GPS points.



Image to Image Geocorrection


Image to Image correction involves matching the coordinate systems or
column and row systems of two digital images with one image acting as a
reference image and the other as the image to be rectified.


Spatial interpolation
: from input position to output position or coordinates.


RST (rotation, scale, and transformation), Polynomial, Triangulation


Root Mean Square Error (RMS):

The RMS is the error term used to
determine the accuracy of the transformation from one system to another. It
is the difference between the desired output coordinate for a GCP and the
actual.


Intensity (or pixel value) interpolation (also called resampling):

The process of
extrapolating data values to a new grid, and is the step in rectifying an image that
calculates pixel values for the rectified grid from the original data grid.


Nearest neighbor, Bilinear, Cubic


Ground control point



Geometric distortions introduced by sensor system
attitude

(roll, pitch,
and yaw) and/or
altitude

changes can be corrected using ground control
points and appropriate mathematical models. A
ground control point

(GCP)

is a location on the surface of the Earth (e.g., a road intersection)
that can be identified on the imagery and located accurately on a map. The
image analyst must be able to obtain two distinct sets of coordinates
associated with each GCP:




image coordinates

specified in
i

rows and
j

columns, and



map coordinates

(e.g.,
x,

y

measured in degrees of latitude and longitude, feet in
a state plane coordinate system, or meters in a Universal Transverse Mercator
projection).



The paired coordinates (
i,

j

and
x,

y
) from many GCPs (e.g., 20) can be
modeled to derive
geometric transformation coefficients
. These
coefficients may be used to geometrically rectify the remote sensor data to
a standard datum and map projection.


a) U.S. Geological Survey 7.5
-
minute 1:24,000
-
scale topographic map of Charleston, SC, with three
ground control points identified (13, 14, and 16). The GCP map coordinates are measured in meters
easting (
x
) and northing (
y
) in a Universal Transverse Mercator projection. b) Unrectified 11/09/82
Landsat TM band 4 image with the three ground control points identified. The image GCP coordinates
are measured in rows and columns.

2.1

2.2 Image to image


Manuel select GCPs (the same as Image to Map)


Automatic algorithms


Algorithms that directly use image pixel values;


Algorithms that operate in the frequency domain (e.g., the
fast Fourier transform (FFT) based approach used);
see
paper Xie et al. 2003


Algorithms that use low
-
level features such as edges and
corners; and


Algorithms that use high
-
level features such as identified
objects, or relations between features.


FFT
-
based automatic image to
image registration


Translation, rotation and scale in spatial domain have
counterparts

in the frequency
domain.


After FFT transform, the
phase difference

in frequency domain will be directly
related to their shifts in the space domain. So we get
shifts
.


Furthermore, both rotation and scaling can be represented as shifts by Converting
from
rectangular coordination

to
log
-
polar coordination
. So we can also get the
rotation and scale.

The Main IDL Codes

Xie et al. 2003

ENVI Main Menus

We added these new submenus

Main Menu
of ENVI


TM Band 7

July 12, 1997

UTM, Zone 13N

TM band 7

June 26, 1991

State Plane, TX Center 4203

This is the new interface

for automatic image

registration


TM Band 7

July 12, 1997

UTM, Zone 13N


TM band 7

June 26, 1991

UTM, Zone 13N

Registered Image

2.3 Spatial Interpolation


RST

(rotation, scale, and transformation or shift) good for image no local
geometric distortion, all areas of the image have the same rotation, scale, and
shift. The FFT
-
based method presented early belongs to this type. If there is
local distortion, polynomial and triangulation are needed.



Polynomial

equations be fit to the GCP data using least
-
squares criteria to
model the corrections directly in the image domain without explicitly
identifying the source of the distortion. Depending on the distortion in the
imagery, the number of GCPs used, and the degree of topographic relief
displacement in the area,
higher
-
order polynomial equations

may be required
to geometrically correct the data. The
order

of the rectification is simply the
highest exponent used in the polynomial.



Triangulation
constructs a Delaunay triangulation of a planar set of points.
Delaunay triangulations are very useful for the interpolation, analysis, and
visual display of irregularly
-
gridded data. In most applications, after the
irregularly gridded data points have been triangulated, the function TRIGRID is
invoked to interpolate surface values to a regular grid. Since
Delaunay
triangulations

have the property that the circumcircle of any triangle in the
triangulation contains no other vertices in its interior, interpolated values are
only computed from nearby points.


Concept of how different
-
order transformations fit a
hypothetical surface
illustrated in cross
-
section.


One dimension:


a)
Original

observations.

b)
First
-
order linear


transformation

fits a


plane to the data.

c)
Second
-
order quadratic



fit.

d)
Third
-
order cubic

fit.

Polynomial interpolation for
image (2D)



This type of transformation can model six kinds of distortion in
the remote sensor data, including:



translation

in
x

and

y
,




scale

changes in
x

and

y
,




rotation
, and



skew
.


When all six operations are combined into a single expression it
becomes:





where
x

and
y

are positions in the
output
-
rectified

image or map, and
x


and
y


represent corresponding positions in the original
input

image.

y
b
x
b
b
y
y
a
x
a
a
x










2
1
0
2
1
0
y
b
x
b
b
y
y
a
x
a
a
x
2
1
0
2
1
0
'
'






a) The logic of filling a rectified
output matrix with values from
an unrectified input image matrix
using
input
-
to
-
output

(
forward
)

mapping logic.

b) The logic of filling a rectified
output matrix with values from
an unrectified input image matrix
using
output
-
to
-
input
(
inverse
)

mapping logic and nearest
-
neighbor resampling.


Output
-
to
-
input inverse

mapping
logic is the preferred
methodology because it results in
a rectified output matrix with
values at every pixel location.

y
x
y
y
x
x
)
0349150
.
0
(
)
005576
.
0
(
130162
'
)
005481
.
0
(
034187
.
0
2366
.
382
'










y
b
x
b
b
y
y
a
x
a
a
x
2
1
0
2
1
0
'
'






The goal is to fill a
matrix that is in a
standard map
projection with the
appropriate values
from a non
-
planimetric image.

root
-
mean
-
square error



Using the six coordinate transform coefficients that model distortions in the original
scene, it is possible to use the output
-
to
-
input (inverse) mapping logic to transfer
(relocate) pixel values from the original distorted image
x

,

y


to the grid of the rectified
output image,
x,

y
.
However, before applying the coefficients to create the rectified
output image, it is important to determine how well the six coefficients derived from the
least
-
squares regression of the initial GCPs account for the geometric distortion in the
input image.

The method used most often involves the computation of the
root
-
mean
-
square error

(
RMS
error)

for each of the ground control points.






2
2
orig
orig
error
y
y
x
x
RMS






where:

x
orig

and
y
orig

are the
original

row and column coordinates of the GCP in the
image and
x’

and
y’

are the
computed or estimated

coordinates in the original
image when we utilize the six coefficients. Basically, the closer these paired
values are to one another, the more accurate the algorithm (and its coefficients).

Cont’


There is an iterative process that takes place. First, all of the original
GCPs (e.g., 20 GCPs) are used to compute an initial set of six coefficients
and constants. The root mean squared error (RMSE) associated with each
of these initial 20 GCPs is computed and summed. Then,
the individual
GCPs that contributed the greatest amount of error are determined and
deleted
.

After the first iteration, this might only leave 16 of 20 GCPs. A
new set of coefficients is then computed using the16 GCPs. The process
continues until the RMSE reaches a user
-
specified threshold (e.g.,
<
1
pixel error in the
x
-
direction and
<
1 pixel error in the
y
-
direction
). The
goal is to remove the GCPs that introduce the most error into the multiple
-
regression coefficient computation. When the acceptable threshold is
reached, the final coefficients and constants are used to rectify the input
image to an output image in a standard map projection as previously
discussed.


2.4 Pixel value interpolation



Intensity (pixel value) interpolation

involves the extraction of a pixel
value from an
x

,

y


location in the original (distorted) input image and its
relocation to the appropriate
x,

y

coordinate location in the rectified output
image. This
pixel
-
filling logic

is used to produce the output image line by
line, column by column. Most of the time the
x


and
y


coordinates to be
sampled in the input image are floating point numbers (i.e., they are not
integers). For example, in the Figure we see that pixel 5,

4 (
x,

y
) in the
output image is to be filled with the value from coordinates 2.4,

2.7
(
x

,

y


) in the original input image. When this occurs, there are several
methods of
pixel value
interpolation

that can be applied, including:




nearest neighbor,



bilinear interpolation,
and




cubic convolution.




The practice is commonly referred to as
resampling
.


Nearest Neighbor

The Pixel value closest to the predicted
x’
,

y’

coordinate is
assigned to the output
x, y

coordinate.

R.D.Ramsey


ADVANTAGES:


Output values are the original input values. Other methods of resampling tend to
average surrounding values. This may be an important consideration when
discriminating between vegetation types or locating boundaries.


Since original data are retained, this method is recommended before classification.


Easy to compute and therefore fastest to use.


DISADVANTAGES:


Produces a choppy, "stair
-
stepped" effect. The image has a rough appearance relative
to the original unrectified data.


Data values may be
lost
, while other values may be
duplicate
d. Figure shows an
input file (orange) with a yellow output file superimposed. Input values closest to the
center of each output cell are sent to the output file to the right.
Notice that values 13
and 22 are lost while values 14 and 24 are duplicated


Original

After linear interpolation

Bilinear


Assigns output pixel values by interpolating brightness values in two orthogonal
direction in the input image. It basically fits a plane to the
4

pixel values nearest to
the desired position (
x’
,
y’
) and then computes a new brightness value based on the
weighted distances to these points. For example, the distances from the requested
(
x’
,
y’
) position at 2.4, 2.7 in the input image to the closest four input pixel
coordinates (2,2; 3,2; 2,3;3,3) are computed . Also, the closer a pixel is to the
desired x’,y’ location, the more weight it will have in the final computation of the
average.


ADVANTAGES:

Stair
-
step effect caused by the
nearest neighbor approach is
reduced. Image looks smooth.


DISADVANTAGES:

Alters original data and
reduces contrast by averaging
neighboring values together.

Is computationally more
expensive than nearest
neighbor.


Dark edge by averaging zero value outside

Original

After Bilinear interpolation

See the FFT
-
based
method has the same
logic

Cubic


Assigns values to output pixels in much the same manner as
bilinear interpolation, except that the weighted values of
16

pixels surrounding the location of the desired
x’, y’

pixel are
used to determine the value of the output pixel.



ADVANTAGES:

Stair
-
step effect caused by
the nearest neighbor
approach is reduced. Image
looks smooth.


DISADVANTAGES:

Alters original data and
reduces contrast by
averaging neighboring
values together.



Is computationally more
expensive than nearest
neighbor or bilinear
interpolation

Original

Dark edge by averaging zero value outside

After Cubic interpolation

3. Image mosaicking


Mosaicking is the process of combining multiple images into a single
seamless composite image.


It is possible to mosaic unrectified individual frames or flight lines of
remotely sensed data.


It is more common to mosaic multiple images that have already been
rectified to a standard map projection and datum


One of the images to be mosaicked is designated as the
base image
. Two
adjacent images normally overlap a certain amount (e.g., 20% to 30%).


A representative
overlapping region

is identified. This area in the base
image is contrast stretched according to user specifications. The histogram
of this geographic area in the base image is extracted. The histogram from
the base image is then applied to image 2 using a
histogram
-
matching
algorithm
. This causes the two images to have approximately the same
grayscale characteristics

Cont’



It is possible to have the pixel brightness
values in one scene simply dominate the
pixel values in the overlapping scene.
Unfortunately, this can result in noticeable
seams in the final mosaic. Therefore, it is
common to blend the seams between
mosaicked images using
feathering
.


Some digital image processing systems
allow the user to specific a
feathering
buffer distance

(e.g., 200 pixels) wherein
0% of the base image is used in the
blending at the edge and 100% of image 2
is used to make the output image.


At the
specified distance

(e.g., 200 pixels)
in from the edge, 100% of the base image
is used to make the output image and 0% of
image 2 is used. At 100 pixels in from the
edge, 50% of each image is used to make
the output file.


Image seamless mosaic

33/38

32/38

31/38

32/39

31/39

Cutline

Symbol

N

Histogram Matching

33/38

32/38

31/38

32/39

31/39

Place Images In a Particular Order

ETM+ 742

fused with pan

(Sept. and Oct.

1999)


Resolution:15m

Size: 2.5 GB

N

112 miles = 180 km